Asymptotic estimates for a two dimensional problem with polynomial nonlinearities

In this paper we give asymptotic estimates of the least energy solution Up of the functional J(u) = ∫Ω |∇u|2 constrained on the manifold ∫Ω |u|p+1 = 1 as p goes to infinity. Here Ω is a smooth bounded domain of ℝ 2. Among other results we give a positive answer to a question raised by Chen, Ni, and Zhou (2000) by showing that lim p→∞ ||up||∞ = √e